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Register a new userReasoning about call-by-value: a missing result in the history of Hoares logic
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Added by
Krzysztof R. Apt
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We provide a sound and relatively complete Hoare-like proof system for reasoning about partial correctness of recursive procedures in presence of local variables and the call-by-value parameter mechanism, and in which the correctness proofs are linear in the length of the program. We argue that in spite of the fact that Hoare-like proof systems for recursive procedures were intensively studied, no such proof system has been proposed in the literature.
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This paper explores two topics at once: the use of denotational semantics to bound the evaluation length of functional programs, and the semantics of strong (that is, possibly under abstractions) call-by-value evaluation. About the first, we analyze de Carvalhos seminal use of relational semantics for bounding the evaluation length of lambda-terms, starting from the presentation of the semantics as an intersection types system. We focus on the part of his work which is usually neglected in its many recent adaptations, despite being probably the conceptually deeper one: how to transfer the bounding power from the type system to the relational semantics itself. We dissect this result and re-understand it via the isolation of a simpler size representation property. About the second, we use relational semantics to develop a semantical study of strong call-by-value evaluation, which is both a delicate and neglected topic. We give a semantic characterization of terms normalizable with respect to strong evaluation, providing in particular the first result of adequacy with respect to strong call-by-value. Moreover, we extract bounds about strong evaluation from both the type systems and the relational semantics. Essentially, we use strong call-by-value to revisit de Carvalhos semantic bounds, and de Carvalhos technique to provide semantical foundations for strong call-by-value.
In each variant of the lambda-calculus, factorization and normalization are two key-properties that show how results are computed. Instead of proving factorization/normalization for the call-by-name (CbN) and call-by-value (CbV) variants separately, we prove them only once, for the bang calculus (an extension of the lambda-calculus inspired by linear logic and subsuming CbN and CbV), and then we transfer the result via translations, obtaining factorization/normalization for CbN and CbV. The approach is robust: it still holds when extending the calculi with operators and extra rules to model some additional computational features.
Most modern (classical) programming languages support recursion. Recursion has also been successfully applied to the design of several quantum algorithms and introduced in a couple of quantum programming languages. So, it can be expected that recursion will become one of the fundamental paradigms of quantum programming. Several program logics have been developed for verification of quantum while-programs. However, there are as yet no general methods for reasoning about (mutual) recursive procedures and ancilla quantum data structure in quantum computing (with measurement). We fill the gap in this paper by proposing a parameterized quantum assertion logic and, based on which, designing a quantum Hoare logic for verifying parameterized recursive quantum programs with ancilla data and probabilistic control. The quantum Hoare logic can be used to prove partial, total, and even probabilistic correctness (by reducing to total correctness) of those quantum programs. In particular, two counterexamples for illustrating incompleteness of non-parameterized assertions in verifying recursive procedures, and, one counterexample for showing the failure of reasoning with exact probabilities based on partial correctness, are constructed. The effectiveness of our logic is shown by three main examples -- recursive quantum Markov chain (with probabilistic control), fixed-point Grovers search, and recursive quantum Fourier sampling.
We present a system called Adelfa that provides mechanized support for reasoning about specifications developed in the Edinburgh Logical Framework or LF. Underlying Adelfa is a new logic named L_LF. Typing judgements in LF are represented by atomic formulas in L_LF and quantification is permitted over contexts and terms that appear in such formulas. Contexts, which constitute type assignments to uniquely named variables that are modelled using the technical device of nominal constants, are characterized in L_LF by context schemas that describe their inductive structure. We present these formulas and an associated semantics before sketching a proof system for constructing arguments that are sound with respect to the semantics. We then outline the realization of this proof system in Adelfa and illustrate its use through a few example proof developments. We conclude the paper by relating Adelfa to existing systems for reasoning about LF specifications.
We present a history of Hoares logic.
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